GEOMETRY OF SYMMETRIC POWERS OF COMPLEX DOMAINS Christopher Grow A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts Department of Mathematics Central Michigan University Mount Pleasant, Michigan April 2017 ABSTRACT GEOMETRY OF SYMMETRIC POWERS OF COMPLEX DOMAINS by Christopher Grow m Given a complex manifold X, its m-fold symmetric power XSym is the collection of all unordered m-tuples of elements of the complex manifold. Such symmetric products arise naturally in complex analysis, for example as the set of solutions of a polynomial equation. The symmetric powers inherit a complex structure, so that it is possible to define holomorphic functions and maps on them. In this work, our goal is to understand aspects m of the complex geometry of the symmetric powers (Bs)Sym, where Bs is the open unit ball in s m the complex vector space C , by classifying the proper holomorphic maps from (Bs)Sym to itself. These proper holomorphic self-maps are distinguished by the fact that they carry the boundary of a space to itself, and are a natural generalization of the notion of biholomorphic automorphisms. In the case when s = 1, the m-th symmetric power of the unit disc in C is known traditionally as the m-dimensional symmetrized polydisc, and has been intensively studied in view of its importance in operator theory and control engineering. The symmetrized polydisc can be thought of as an open subset of Cm. Edigarian and Zwonek (2005) determined the proper holomorphic self-maps of the symmetrized polydisc, and showed that these self-maps arise from proper holomorphic self-maps of the unit disc. The main result of this thesis is a generalization of the result of Edigarian and Zwonek m to the symmetric power (Bs)Sym of the unit ball Bs, where s ≥ 2. As a first step, we show m N that (Bs)Sym can be identified with an open subset of an affine algebraic variety in C for a large integer N depending on m and s. This is done by using a symmetric analog of the classical Segre Embedding of a product of projective varieties in a higher dimensional m projective space. This gives an elementary way to endow (Bs)Sym with a complex structure. m Then we reduce the problem of classifying proper holomorphic self-maps of (Bs)Sym to a ii certain lifting problem, the obstruction to solving which lies in the fundamental group of m m a certain open subset of (Bs) , obtained by removing an analytic set from (Bs) . Using a transversality argument, we show that this obstruction in fact vanishes, thus proving the m result. As a consequence, we find that each proper holomorphic self-map of (Bs)Sym is an automorphism induced by an automorphism of Bs. iii TABLE OF CONTENTS CHAPTER I. INTRODUCTION . 1 I.1. The Symmetrized Polydisc . .1 I.2. Holomorphic Maps and the Theorem of Edigarian and Zwonek . .2 I.3. Proper Holomorphic Mappings of Higher Dimensional Symmetric Powers .4 I.4. Topological Properties of Proper Maps . .8 I.5. Some Examples of Proper Holomorphic Maps . 10 II. ANALYTIC SETS AND PROPER MAPS . 13 II.1. Analytic Subsets of Complex Manifolds . 13 II.2. Proper Holomorphic Maps . 15 III. SYMMETRIC POWERS . 21 III.1. Definitions and Preliminaries . 21 III.2. Topological Symmetric Powers . 22 III.3. Algebraic Embedding of Symmetric Powers of Projective Spaces . 25 III.4. Embedding of Symmetric Powers of Affine Spaces and of Balls . 31 III.5. Complex Symmetric Powers . 34 III.6. Symmetric Powers as Complex Spaces . 37 IV. PROPER HOLOMORPHIC SELF-MAPS OF SYMMETRIC POWERS OF BALLS 39 IV.1. Parametrized Analytic Sets . 39 IV.2. Some Results of Remmert-Stein Theory . 40 IV.3. Results of Edigarian and Zwonek . 44 IV.4. Mappings from Cartesian to Symmetric Powers . 47 IV.5. End of the Proof of Theorem 3 . 50 REFERENCES . 52 iv CHAPTER I INTRODUCTION I.1. The Symmetrized Polydisc Let D ⊂ C be the unit disc: D = fz 2 Cj jzj < 1g. The subset Σ2D of C2 given by 2 2 Σ D = f(z + w; zw) 2 C jz; w 2 Dg is called the Symmetrized Bidisc. It can be shown that this is a domain, by which we mean an open connected set, which is pseudoconvex but has nonsmooth boundary. This domain first emerged in the study of certain problems in Electrical Engineering (see [1]). However, the symmetrized bidisc is a natural object to consider in complex analysis of several variables. Consider, for example, a quadratic equation in t with complex coefficients s and p: t2 − st + p = 0 such that both roots z; ! of this equation lie in the unit disc. Then clearly the coefficients (s; p) lie in Σ2D. It is possible to generalize this construction, and define the m-dimensional Sym- m th metrized polydisc Σ D in the following way. For 1 ≤ k ≤ m, denote by σk the k elementary symmetric polynomial in m variables. These polynomials are defined to be σ1(z1; : : : zm) = Pm P j=1 zj, σ2(z1; : : : ; zm) = j<k zjzk and so on, with σm(z1; : : : ; zm) = z1z2 : : : zm, i.e., the kth elementary symmetric polynomial is the sum over all products of k distinct z1; : : : ; zm. Then, m m Σ D = f(σ1(z); σ2(z); : : : ; σm(z)) 2 C jz = (z1; : : : ; zm); where each zj 2 Dg: m m As before, a point (s1; : : : ; sm) 2 C lies in Σ D if and only if all the roots of the polynomial equation m m−1 m−2 m t − s1t + s2t − · · · + (−1) sm = 0 1 lie in D. Since the appearance of [1], the domains ΣmD have been a subject of much study from the complex analytic point of view (see e.g. [2, 3, 4, 11, 12, 20, 14, 15, 8]). Perhaps the most remarkable property of these domains is that though they are proper holomorphic images of convex domains, the theorem of Lempert on the equality of the Kobayashi and Carathedory metrics on convex domains does not extend to them (see [11]). I.2. Holomorphic Maps and the Theorem of Edigarian and Zwonek One of the avenues of this work has been to explore mapping properties and symme- tries of the domain ΣmD. Such questions regarding holomorphic mappings are very natural and important in complex analysis. Suppose U and V are open subsets of Cn. If we have a bijective holomorphic map f : U ! V then f −1 is also holomorphic. In this case, we call f a biholomorphism. One of the famous theorems of mathematics is the Riemann Mapping The- orem which states that each simply connected proper subdomain of C is biholomorphic to the unit disc D. It is also well-known how to determine the group Aut(D) of all automorphisms of D (i.e. biholomorphisms from D to itself). For domains in Cn with n ≥ 2, the study of holomorphic maps is much more compli- cated, and there is no higher-dimensional analog of the general theory of conformal mapping. In particular, there is no simple analog of the Riemann Mapping Theorem: not all open, simply-connected proper subsets of Cn are biholomorphic to the unit ball n 2 2 1 Bn := fz 2 C : jzj = (jz1j + ··· + jznj ) 2 < 1g (I.1) 2 when n ≥ 2. The simplest counter-example is the domain D = D × D, where D = B1, the 2 unit disc. Although D is in fact homeomorphic to B2, there is no biholomorphism between these domains, a fact known to Poincar´e. We give a proof of this result based on a more general theorem of Remmert and Stein (see [21]) in Chapter IV. In spite of the difficulties in studying general biholomorphic mappings of several 2 variables outlined above, it is still sometimes possible to characterize mappings between special domains. In this thesis, which is motivated by the results of Jarnicki, Pflug, Edigarian, and Zwonek, which we describe below, we study the structure of interesting holomorphic maps between some higher dimensional analogs of the symmetrized polydisc. We begin by summarizing the known results about holomorphic maps between symmetrized polydiscs, on which our work is based. The first step in this direction is [20], in which Jarncki and Pflug explicitly determined the group of automorphisms of the symmetrized polydisc. We do not state this result sepa- rately, since it is a special case of the results of Edigarian and Zwonek on proper holomorphic maps of these domains, which we will now describe. A proper map is a weakening of the notion of a homeomorphism between topological spaces. For f to be a proper map, instead of requiring that f is bijective and has a contin- uous inverse, we require that the inverse image of a compact set is compact. Clearly, each homeomorphism is a proper map, but not every proper map is a homeomorphism. Here, as well as in subsequent chapters, by a map, we mean a continuous function between topological spaces. In complex analysis of one variable, proper holomorphic maps arise naturally as \branched coverings". A typical example of a proper holomorphic map is given by the map f from the unit disc D to itself given by f(z) = zd where d is a positive integer. Notice that (1) the map is proper, since the inverse image of any closed subdisc fjzj ≤ rg of D (where 0 < r < 1) is a closed subdisc of D, (2) the map f takes the boundary of D to itself, and (3) on D n f0g, f is a covering map of d sheets.